The renowned mathematician David Blackwell died on July 8 at the age of 91.
I first came across Blackwell's name in a widely-cited paper by Kalai and Lehrer on learning in repeated games. Kalai and Lehrer identified conditions under which players with different initial subjective beliefs about each others' strategies will nevertheless converge to behavior that approximates a Nash equilibrium of the repeated game. In establishing this, the authors relied heavily on the Blackwell-Dubins Theorem:
Our proof of the convergence to playing an ε-Nash equilibrium is divided into three steps. The first establishes a general self-correcting property of Bayesian updating. This is a modified version of the seminal Blackwell and Dubins' (1962) result about merging of opinions... When applied to our model, the self-correcting property shows that the probability distributions describing the players' beliefs about the future play of the game must converge to the true distribution. In other words, the beliefs and the real play become realization equivalent.
While Blackwell's work is familiar in economics largely through this result, he is also known for the Rao-Blackwell Theorem and his book (with Meyer Girshick) on the Theory of Games and Statistical Decisions. [Update: A much fuller discussion of his influence and contributions may be found here.]
Blackwell earned his doctorate in mathematics at the age of 22 from the University of Illinois, where his thesis adviser was Joseph Doob. He then spent a year at the Institute for Advanced Study in Princeton, where his cohort included Shizuo Kakutani, Paul Halmos, Leonard Savage, and Alfred Tarski. He was elected to the National Academy of Sciences in 1965 and was the sole recipient of the John von Neumann Theory Prize in 1979 (sandwiched between Nash and Lemke in 1978 and Gale, Kuhn and Tucker in 1980).
While his accomplishments are stellar and many, it is also worth contemplating the many slights that Blackwell had to endure over the course of his career:
Blackwell was appointed a Postdoctoral Fellow at the Institute for Advanced Study from 1941 for a year. At that time, members of the Institute were automatically officially made visiting fellows of Princeton University, and thus Blackwell was listed in its bulletin as such. This caused considerable ruckus as there had never been a black student, much less faculty fellow, at the University... The president of Princeton wrote the director of the Institute that the Institute was abusing the University's hospitality by admitting a black... Colleagues in Princeton wished to extend Blackwell's appointment at the institute. However, the president of Princeton organized a great protestation... When it was time to leave the institute, Blackwell knew no white schools would hire him, and he applied to all 105 Black schools in the country. After instructorships at Southern University and Clark College, Dr. Blackwell joined the faculty of Howard University from 1944 as an instructor... In three years, Blackwell had risen to the rank of Full Professor and Chairman.
Blackwell eventually moved to Berkeley in 1954 (after having previously been denied a position there due to "racial objections"). He became the first black professor to be tenured there, chaired the department of statistics from 1957 to 1961, and remained at the University until his retirement in 1988.
It takes a particular kind of strength to manage such a productive research career while tolerating the stresses and strains of personal insult, and carrying the aspirations of so many on one's shoulders. Blackwell was more than a brilliant mathematician, he was also a human being of extraordinary personal fortitude.
I am currently in Bogotá co-teaching a course with Glenn Loury at the (very impressive) Universidad de Los Andes. I am grateful to Glenn for bringing to my attention the news that Blackwell had recently passed away.
Update (7/19). Jeff Ely has linked to two other posts in appreciation of Blackwell: Eran Shmaya focuses on his work while Jesús Fernández-Villaverde writes (in Spanish) about his life. Both are well worth reading. Here's an extract from Eran's wonderful post:
We game theorists know Blackwell for several seminal contributions. Blackwell’s approachability theorem is at the heart of Aumann and Maschler’s result about repeated games with incomplete information... Blackwell’s theory of comparison of experiments has been influential in the game-theoretic study of value of information... Another seminal contribution of Blackwell, together with Lester Dubins, is the theorem about merging of opinions, which is the major tool in the Ehuds’ theory of Bayesian learning in repeated games. And then there are his contributions to the theory of infinite games with Borel payoffs (now known as Blackwell games) and Blackwell and Fergurson’s solution to the Big Match game.Kevin Bryan has also put up a nice post on Blackwell.
One conspicuous aspect of many of Blackwell’s awesome papers is that they are extremely short — often a couple of pages long. He had an amazing ability to prove theorems in the right way, and he wrote with eloquence and clarity. He is the only writer I know who uses the 'as the reader can verify' trick productively, exactly at those occasions when the reader will indeed find it easier to convince himself in the validity of an assertion than to read a formal proof of it. It is very rare that I succeed in reading proofs in papers that were written dozens of years ago: Notations and perspectives change, and important results are usually reproduced in clearer way over the years. But Blackwell’s papers are still the best place to read the proofs of his theorems...
I hope I am not forcing my own agenda on Blackwell’s research when I say that for him game and decision theory were a tool to study conceptual questions about the meaning of probability and information. At any rate, he was clearly interested in these questions... I hope the game theory society will find a way to celebrate Blackwell’s contribution to our community.
Update (7/20). Stergios (in a comment on this post) observes that "Blackwell also made fundamental contributions to the theory of stochastic processes" and that his "renewal theorem is taught in any doctoral level course on stochastic models." And Glenn Loury has emailed me a link to a paper by Jacques Crémer "nicely expositing one of Blackwell's more influential results in statistical decision theory."
Andrew Gelman (and his commenters) have more. And Anandaswarup Gadde links to a wonderful profile from about a year ago:
Doob’s foundational work would help broaden the field of mathematics to a dizzying array of uses in science, economics and technology. So it came as no surprise when in 1942, Jerzy Neyman of the University of California at Berkeley asked if Doob were interested in going West.
“No, I cannot come, but I have some good students, and Blackwell is the best,” he replied.
“But of course he’s black,” Doob continued, “and in spite of the fact that we are in a war that’s advancing the cause of democracy, it may not have spread throughout our own land.”
The quote, repeated in the book “Mathematical People,” says a lot about the times and even more about David H. Blackwell... who started as an Illinois undergraduate in 1935 and finished with a doctoral degree six years later, all accomplished at a time when residence halls were whites-only, and approximately 100 blacks were included in the student body of nearly 12,000.
What would be the odds of the son of a railroad worker from Centralia – whose parents did not complete high school and whose Depression-era teaching prospects were limited to segregated schools – becoming one of the top theoretical mathematicians (black or white) in the world?
Almost too hard to compute...
After earning a UI doctoral degree in mathematics in 1941 at the age of 22, Blackwell completed a year at the Institute for Advanced Study in Princeton, N.J., where he worked with, among others, John von Neumann, father of modern game theory.
Berkeley’s Jerzy Neyman – who had been unable to persuade Doob to join his department – wanted to offer Blackwell a position but appeared to have come up against a deal-breaker.Read the whole thing.
In an oral history interview at Berkeley, Blackwell, now 90 years old and in “fair” health, recalled what he learned years later – that the Texan wife of the department head told her husband she “was not going to have that darky in her house.”
The job offer never came.
Blackwell focused his efforts instead on realistic career aspirations for a person of color at the time. In 1942 he applied to 105 historically black colleges, received three offers and eventually landed at Howard University in Washington, D.C., in 1944, where he remained for 10 years...
Back at Berkeley, Neyman had never forgotten Blackwell and finally hired him in 1954, where he would stay for the remainder of his career.